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Theorem brabga 4237
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brabga.2  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabga  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)    V( x, y)    W( x, y)

Proof of Theorem brabga
StepHypRef Expression
1 df-br 3984 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabga.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2320 . . 3  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
41, 3bitri 242 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
5 opelopabga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
65opelopabga 4236 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
74, 6syl5bb 250 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   <.cop 3603   class class class wbr 3983   {copab 4036
This theorem is referenced by:  braba  4240  brabg  4242  epelg  4264  brcog  4824  fmptco  5611  ofrfval  6006  wemaplem1  7215  oemapval  7339  wemapwe  7354  fpwwe2lem2  8208  fpwwelem  8221  clim  11919  rlim  11920  vdwmc  12973  isstruct2  13105  brssc  13639  isfunc  13686  isfull  13732  isfth  13736  ipole  14209  eqgval  14614  frgpuplem  15029  dvdsr  15376  ulmval  19707  hlimi  21713  isumgra  23225  iseupa  23239  isside1  25518  islindf  26635
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038
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